Confidence Intervals for a Population Mean ($\sigma$ Unknown)

Confidence Interval: ${\displaystyle \overline{x}\pm z_{\alpha /2}\cdot \frac{\sigma }{\sqrt{n}}}$

Extra uncertainty means the standard normal curve no longer gives the right critical value.

It comes from something called the Student's $t$ distribution.

Why the extra spread?

We are estimating $\sigma$ with $s$, which introduces uncertainty. To maintain the same confidence level, we need a wider interval. The $t$ distribution builds this in.

Why is there more than one?

Different sample sizes have different amounts of uncertainty. Each needs its own distribution.

We distinguish them by a quantity related to the sample size called degrees of freedom.

The number of degrees of freedom is $n-1$.

1 Degree of Freedom

4 Degrees of Freedom

10 Degrees of Freedom

Degrees of freedom $=n-1$

Fewer degrees of freedom → more uncertainty → wider distribution

More degrees of freedom → $s$ is close to $\sigma$ → nearly normal

${\displaystyle \overline{x}\pm t_{\alpha /2}\cdot \frac{s}{\sqrt{n}}}$

$t_{\alpha /2}$ is noticeably larger than $z_{\alpha /2}$ when $n$ is small, but gets closer to it as $n$ grows.

When $\sigma$ is known:

CLT → Sampling distribution of $\overline{x}$ is normal

When $\sigma$ is not known:

Same CLT conditions apply → Use $t$

Caveat:

When the sample size is small, we must check whether the population can be assumed to be normal. Draw a dotplot and look for skewness or outliers.

We are 90% confident that the mean sleep time is between 5.99 and 7.72 hours.

claim: 310 miles

$\overline{x}=298.4$, $s=24.1$, $n=70$

We are 95% confident that the mean driving range is between 292.7 and 304.1 miles.

Because 310 is not in the interval, the confidence interval does contradict the manufacturer's claim.

claim: less than 60 minutes

$\overline{x}=54.2$, $s=17.6$, $n=100$

We are 90% confident that the mean waiting time is between 51.3 and 57.1 minutes.

Because all values in the confidence interval are less than one hour (60 minutes), the confidence interval does not contradict the hospital's claim.

A coffee shop owner wants to estimate the mean tip amount for a given day. A random sample of 9 tips yielded the following amounts (in dollars):

0.750.251.000.820.801.5010.000.501.50

Construct a 95% confidence interval for the mean tip amount.

(a)Yes

(b)No

(c)Yes

(d)No: a voluntary response sample

The Question:

Among people hospitalized with COVID-19, do vaccinated patients have better outcomes than unvaccinated patients?

The Data:

What Was Measured:

The confidence intervals do not overlap in either case.

Vaccinated Patients:

Keep in Mind:

This study only looked at patients already hospitalized. Vaccination also reduces the chance of infection, severe symptoms, and hospitalization in the first place.